Theorem sprel 48274

Description: An element of the set of all unordered pairs over a given set 𝑉 is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)

Assertion

Ref	Expression
sprel	⊢ (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏})

Distinct variable groups:   𝑉,𝑎,𝑏   𝑋,𝑎,𝑏

Proof of Theorem sprel

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6923	. 2 ⊢ (𝑋 ∈ (Pairs‘𝑉) → 𝑉 ∈ V)
2		sprvalpw 48270	. . . 4 ⊢ (𝑉 ∈ V → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})
3	2	eleq2d 2852	. . 3 ⊢ (𝑉 ∈ V → (𝑋 ∈ (Pairs‘𝑉) ↔ 𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}))
4		eqeq1 2770	. . . . . 6 ⊢ (𝑝 = 𝑋 → (𝑝 = {𝑎, 𝑏} ↔ 𝑋 = {𝑎, 𝑏}))
5	4	2rexbidv 3233	. . . . 5 ⊢ (𝑝 = 𝑋 → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏}))
6	5	elrab 3653	. . . 4 ⊢ (𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ↔ (𝑋 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏}))
7	6	simprbi 503	. . 3 ⊢ (𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏})
8	3, 7	biimtrdi 256	. 2 ⊢ (𝑉 ∈ V → (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏}))
9	1, 8	mpcom 39	1 ⊢ (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ∃wrex 3092  {crab 3419  Vcvv 3458  𝒫 cpw 4567  {cpr 4596  ‘cfv 6543  Pairscspr 48267

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-spr 48268

This theorem is used by:  prssspr  48275  prsprel  48277  reupr  48312